Duration and convexity

2 free worked examples

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1. Exam FMDuration and convexityExam level

   Find the Macaulay duration of a 5-year 1,000 par bond with 6% annual coupons at a yield of 6%.

   1. A$4.2124$
   2. B$4.4651$
   3. C$4.7500$
   4. D$5.0000$
   5. E$5.3000$

   Solution

   1. Macaulay duration is the present-value-weighted average time to the cash flows.
   2. Cash flows are 60 at times 1 to 4 and 1,060 at time 5; at a 6% yield the price is exactly 1,000.
   3. $D = \dfrac{\sum t\,C_t v^{t}}{1{,}000}$ with $v = 1/1.06$.
   4. $= 4.4651$ years, less than the 5-year term because four coupons arrive first.

   Trap. Weighting by the cash flows themselves rather than by their present values.

2. Exam FMDuration and convexityExam level

   For the 5-year 6% par bond at a 6% yield, find the Macaulay convexity.

   1. A$19.9500$
   2. B$21.2863$
   3. C$22.9187$
   4. D$24.0000$
   5. E$25.5000$

   Solution

   1. Macaulay convexity is $\dfrac{\sum t^{2}C_t v^{t}}{P}$ - the second PV-weighted moment of the payment times.
   2. Working that sum over the five cash flows at a 6% yield and dividing by the price of 1,000 gives 21.2863.
   3. The MODIFIED convexity $\dfrac{\sum t(t+1)C_tv^{t+2}}{P} = 22.9187$ is a different quantity.
   4. Only the modified version belongs in the price-change estimate; mixing them is the standard error here.

   Trap. Using Macaulay convexity in the second-order price approximation.